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Connections and dynamical trajectories in generalised Newton-Cartan gravity I. An intrinsic view

Xavier Bekaert, Kevin Morand

TL;DR

The work develops an intrinsic framework for generalized Newton–Cartan gravity by analyzing how nonrelativistic metric data (absolute clock and rulers) constrain compatible connections, including both torsionfree and torsional cases. It reveals that, unlike relativity, there is no canonical origin for metric-compatible connections in the nonrelativistic setting; instead, a privileged class of torsional Newton–Cartan geometries provides a natural origin and an affine-space structure governed by 2-forms and Milne boosts. A Lagrangian structure is introduced to single out a unique connection, and the analysis is extended to torsional cases with a covariantly exact torsion component, together with an ambient Bargmann perspective linking nonrelativistic and relativistic geometries. The results illuminate how dynamics in nonrelativistic spacetimes can be encoded as geodesic motion under generalized connections, with potential applications to condensed matter, holography, and ambient formulations of Newtonian trajectories.

Abstract

The "metric" structure of nonrelativistic spacetimes consists of a one-form (the absolute clock) whose kernel is endowed with a positive-definite metric. Contrarily to the relativistic case, the metric structure and the torsion do not determine a unique Galilean (i.e. compatible) connection. This subtlety is intimately related to the fact that the timelike part of the torsion is proportional to the exterior derivative of the absolute clock. When the latter is not closed, torsionfreeness and metric-compatibility are thus mutually exclusive. We will explore generalisations of Galilean connections along the two corresponding alternative roads in a series of papers. In the present one, we focus on compatible connections and investigate the equivalence problem (i.e. the search for the necessary data allowing to uniquely determine connections) in the torsionfree and torsional cases. More precisely, we characterise the affine structure of the spaces of such connections and display the associated model vector spaces. In contrast with the relativistic case, the metric structure does not single out a privileged origin for the space of metric-compatible connections. In our construction, the role of the Levi-Civita connection is played by a whole class of privileged origins, the so-called torsional Newton-Cartan (TNC) geometries recently investigated in the literature. Finally, we discuss a generalisation of Newtonian connections to the torsional case.

Connections and dynamical trajectories in generalised Newton-Cartan gravity I. An intrinsic view

TL;DR

The work develops an intrinsic framework for generalized Newton–Cartan gravity by analyzing how nonrelativistic metric data (absolute clock and rulers) constrain compatible connections, including both torsionfree and torsional cases. It reveals that, unlike relativity, there is no canonical origin for metric-compatible connections in the nonrelativistic setting; instead, a privileged class of torsional Newton–Cartan geometries provides a natural origin and an affine-space structure governed by 2-forms and Milne boosts. A Lagrangian structure is introduced to single out a unique connection, and the analysis is extended to torsional cases with a covariantly exact torsion component, together with an ambient Bargmann perspective linking nonrelativistic and relativistic geometries. The results illuminate how dynamics in nonrelativistic spacetimes can be encoded as geodesic motion under generalized connections, with potential applications to condensed matter, holography, and ambient formulations of Newtonian trajectories.

Abstract

The "metric" structure of nonrelativistic spacetimes consists of a one-form (the absolute clock) whose kernel is endowed with a positive-definite metric. Contrarily to the relativistic case, the metric structure and the torsion do not determine a unique Galilean (i.e. compatible) connection. This subtlety is intimately related to the fact that the timelike part of the torsion is proportional to the exterior derivative of the absolute clock. When the latter is not closed, torsionfreeness and metric-compatibility are thus mutually exclusive. We will explore generalisations of Galilean connections along the two corresponding alternative roads in a series of papers. In the present one, we focus on compatible connections and investigate the equivalence problem (i.e. the search for the necessary data allowing to uniquely determine connections) in the torsionfree and torsional cases. More precisely, we characterise the affine structure of the spaces of such connections and display the associated model vector spaces. In contrast with the relativistic case, the metric structure does not single out a privileged origin for the space of metric-compatible connections. In our construction, the role of the Levi-Civita connection is played by a whole class of privileged origins, the so-called torsional Newton-Cartan (TNC) geometries recently investigated in the literature. Finally, we discuss a generalisation of Newtonian connections to the torsional case.

Paper Structure

This paper contains 27 sections, 45 theorems, 114 equations, 7 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Outline
  3. Notations
  4. Notations
  5. Nonrelativistic metric structures
  6. Relativistic structures
  7. Nonrelativistic structures
  8. Observers
  9. Absolute time and spaces
  10. Milne boosts
  11. Torsionfree Galilean connections
  12. Galilean manifolds
  13. Torsionfree Galilean manifolds
  14. Remarks (on Milne invariance and special connections):
  15. Torsionfree Newtonian manifolds
  16. ...and 12 more sections

Key Result

Theorem 2.4

The space of Lorentzian connections compatible with a given Lorentzian structure $\left(\mathscr{M},g\right)$ forms a vector space which is isomorphic to the vector space $\Gamma\left( \wedge^2\, T^*\mathscr{M}\otimes T\mathscr{M}\right)$ of torsion tensors and the origin of which is the Levi-Civita

Figures (7)

  • Figure 1: Kinematical content of general relativity
  • Figure 2: Kinematical content of metrical theories of gravitation
  • Figure 3: Kinematical content of Newton-Cartan theory
  • Figure 4: Foliation of an Aristotelian structure by absolute spaces.
  • Figure 5: Representation change under a Milne boost
  • ...and 2 more figures

Theorems & Definitions (83)

  • Definition 2.1: Riemannian structure
  • Definition 2.2: Lorentzian structure
  • Definition 2.3: Riemannian/Lorentzian manifold
  • Theorem 2.4: Space of metric compatible connections
  • Proposition 2.5
  • Lemma 2.6
  • Proposition 2.7
  • Theorem 2.8: Fundamental Theorem of (pseudo)-Riemannian geometry
  • Definition 2.9: Lorentzian basis
  • Definition 2.10: Absolute clock Dombrowski1964Kunzle1972
  • ...and 73 more